22 C : 231 Design and Analysis of Algorithms Midterm Exam The duration of this exam is one hour and Ffteen minutes. This is a closed-book exam. 1. (10 points) (a) We are given a ±ow network G with vertex set V and a nonnegative integer capacity c ( u, v ) for any edge ( u, v ) ∈ V × V . We are also speciFed a source s ∈ V and a sink t ∈ V . State the deFnition of a ±ow in the network G . DeFne the value of the ±ow. (b) Consider the Fgure on the next page. What is the value of the ±ow that is depicted? (c) Argue that ±ow conservation holds at v 3 . (d) Argue that the ±ow depicted is a maximum ±ow. (e) In the Fgure, what is the capacity of the minimum-capacity s-t cut? 2. (10 points) We are given a sequence S = h s 1 , s 2 , . . . , s n i of n distinct positive in-tegers. The sequence is not necessarily sorted. The problem is to develop an al-gorithm for Fnding the longest increasing subsequence of S . ²or example, if S = h 100 , 10 , 60 , 70 , 20 , 30 , 40 , 80 i

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